Minimal model for higher-order topological insulators and phosphorene

Abstract: A higher order topological insulator is an extended notion of the conventional topological insulator, which belongs to a special class of topological insulators where the conventional bulk-boundary correspondence is not applicable. The bulk topological index may be described by the Wannier center located at a high symmetry point of the crystal. In this paper we propose minimal models for the second-order topological insulator in two dimensions and the third-order topological insulator in three dimensions. They are anisotropic two-band models with two different hopping parameters. The two-dimensional model is known to capture the essential physics of phosphorene near the Fermi level. It has so far been recognized as a trivial insulator due to the absence of topological edge states in nanoribbons. However, we demonstrate the emergence of topological boundary states in zero dimension, i.e., in nanodisks. In particular, the diamond structure exhibits such topological states at two corners, each of which carries a 1/2 fractional charge. We predict that these corner states will be observed in the diamond structure of phosphorene.